Ulams Spiral

FIGURE 2.24 Ulam s spiral showing locations of prime numbers smaller than 100, which tend to be lined along various diagonals. 

FIGURE 2.25 Ulam s spiral for the first 1000 prime numbers. 

Formulas that produce prime numbers, known as generating formulas, have been known for a while. In 1772, Euler found the quadratic polynomial P(n) = n + n + 4l, which, for all values of n = 0,1,2,3. 39 gives prime numbers, 40 primes in total. These are a selection of prime numbers that start with 41,43,47,53,61,71,83. for n = 0 to n = 6 and so on and end with 1601, for n = 39. For n = 40, one obtains for P(n)  

Let us mention two quadratic forms that are related to the Euler polynomial. In 1798, Legendre considered the polynomial  

This formula gives 26 prime numbers, for all n from 0 to 25. 

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The Ulam spiral is a result of Ulam attending an uninteresting lecture, during which he started sketching his prime number spiral. This somewhat unusual format for writing sequences offers a two-dimensional pattern for a one-dimensional mathematical object, here, the sequence of prime numbers [124], In Figure 2.24, we have illustrated Ulam s spiral for prime numbers smaller than 100. 

Before leaving the Ulam spiral, we should mention that there have been a few modifications of Ulam s spiral. For example, R. Sacks constructed the Archimedean spiral by plotting integers uniformly on the spiral, and when composite numbers have been deleted, one obtains what is known as the Sacks prime number spiral [128]. On this spiral, prime numbers that are obtained from Euler s prime number generator x - x+41 are clearly seen on a line approaching the left horizontal axis. There are modifications of the... 

FIGURE 2.26 Patterns of the smallest 190 odd prime numbers on the Ulam spiral constructed on a 24 X 24 Cartesian grid for odd numbers only. 

With the availability of computers, it has been recognized that not only can some problems (which up to that time were beyond the possibility of being numerically solved) be reexamined and solved, but also new problems can emerge, be discovered, considered, and even solved, which were unknown in pre-computer time. Let us briefly mention two such problems that have some novelty, both of which have some connection with chemistry (i) the proof or almost a proof of the Kepler conjecture about dense packing of spheres and (ii) the problan relating to Ulam s spiral. 

We thought that we might add a novelty to the topic of prime number spirals by considering only odd numbers in the construction of the spiral, hi Figure 2.26, we have illustrated a section of a so-modified Ulam s spiral on a 24 x 24 Cartesian grid. 

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